Group on elliptic curve points

Question

Let's we have an elliptic curve (EC). Is it possible to construct group $G$ acting on the points of an EC with this property: if $P$ is a rational point on EC then $G(P)$ is also is a rational point? Of course, I means that $G$ is not reduced to addition of $P$ with itself. In other words, can we construct symmetry group?

Answer 1

If $E/K$ is an elliptic curve over a field $K$, then you can construct the automorphism group of $E$, but it is not terribly exciting... $\text{Aut}(E)$ is a finite group of order dividing $24$, and if $j\neq 0,1728$, then $\text{Aut}(E) = \{\pm 1\}$. See Chapter III.10 of Silverman's "The Arithmetic of Elliptic Curves".

If you restrict yourself to torsion subgroups, however, then there are more exciting automorphism groups. Let $E/K$ be an elliptic curve over a field of characteristic $0$, and let $n\geq 2$. Let $E[n]$ be the $n$-torsion subgroup of $E(\overline{K})$. Then, one can construct $\text{Aut}(E[n])\cong \text{Gal}(K(E[n])/K)$ and this group is a subgroup of $\text{GL}(2,\mathbb{Z}/n\mathbb{Z})$. If $n=p$ is a prime number, $K$ is a number field, and $E/K$ is fixed, then we know that $\text{Gal}(K(E[p])/K)\cong \text{GL}(2,\mathbb{Z}/p\mathbb{Z})$ for all but finitely many primes $p$ (a consequence of the so-called Serre's open image theorem).